Errors in FEA and Understanding Singularities (Beginners’ Guide)
In general, we can decompose errors in FEA – Finite Element Analysis – in three main groups:
- Modeling errors due to simplifications (“We try to model the real world yet are not able to do it 100%.”)
- Discretization errors that arise from the creation of the mesh
- Numerical errors of the solution of the FEA equations
Furthermore, you have to keep in mind that the Finite Element Method, as well as other numerical methods (FVM, FDM, BEM, etc.) are just approximations!
The finite element description is a boundary value problem (BVP) meaning that we have a differential equation with a number of constraints called boundary conditions.
A solution to a BVP is a solution to the differential equation which also satisfies the boundary condition (which I will elaborate in another article in more detail).1
Errors of this type can include:
• wrong geometric description: For instance, we use axial symmetry or rotational symmetry but forget that we have an antisymmetric load
• wrong definition of the material (for example the limit of Poisson’s ratio in isotropic materials)
• wrong definition of the load: trying to simplify complex load states or a number of loads with one load (depends on the case). But often “the difference between the effects of two different but statically equivalent loads becomes very small at sufficiently large distances from load” as Saint-Venant’s Principle expresses
• wrong boundary conditions: forget to fix your model (rigid body motion) or set boundary conditions that may falsify the results that you actually wanted to have
• or a wrong type of analysis
Concerning the analysis types, we have several options:
- Do we have a material linearity or non-linearity?
- Do we have a geometric linearity or non-linearity? Large deformation or strains?
- Contact definition: with friction? Without friction? Master-Slave definition? (For a deeper understanding of Master-Slave definitions see this topic)
- Do we have buckling?
- Do we have a time-dependent problem or can we look at the steady state meaning we assume a quasi-static behavior?
Small Excursus to Singularities
A finite element model will sometimes contain a so-called singularity which means there are points in your model where values tend toward an infinite value.4
Well if you are a newbie in FEA you might assume a singularity is a term derived from a science-fiction movie like Star Trek.
The image below shows you what a singularity is5:
A singularity as shown in the picture above is basically the same in your FEA model where the infinite density and gravity are equivalent to an infinite stress at a sharp corner for example.
Singularities might be confusing because they cause an accuracy problem inside your model which implies a problem of visualization because singularities extend the range of your stresses. This means that smaller stresses look like they are negligible.
But what causes singularities? If you look on the internet, you will find tons of causes and explanations. But summa summarum we can say that boundary conditions are the major cause.
Another predestinated problem which implies singularity problems are of course cracks because it can be seen as a 180° re-entrant corner. Luckily we do not have to study the crack tip every time but can rather focus on the stress intensity factor and use the J-Integral or look at the energy dissipated during fracture (strain energy release rate).
Also, pay attention if you apply a force to a single node! This will give you infinite stresses! In our real world forces are not applied on a single node anyway due to the Saint-Venant’s principle, which tells us that “… the difference between the effects of two different but statically equivalent loads become very small at sufficiently large distances from the load.” I also encountered this problem when I modeled a Hertzian contact in ANSYS and applied my force on a single node on the top of my cylinder. Took me two days to realize my mistake…
- Which type of elements should we use? If we are in 2D: Plane Stress or Plane Strain?
- Which mesh density is accurate enough so that we have a good solution without investing an immense amount of time? ALWAYS START WITH A COARSE MESH! See how the time and accuracy change with an increase in the mesh fineness.
- Which element order should I use? In SimScale you have two possibilities:
- First order tetrahedral elements
- Second order tetrahedral elements
Second order elements may assume either concave or convex shape meaning that they can display better results and deformation by easily mapping to curvilinear geometry. A big advantage when handling with material nonlinearity i.e. elastoplastic or hyperelastic material. But everything comes at a price. The computational effort and time consumption is higher than using first order tetrahedral elements.
Mesh Refinement Methods
h-method: reducing the size of your mesh
p-method: increase of the polynomial order in the element (good for regions with a low-stress gradient)
r-method: relocates the position of a node
Or combinations of the methods mentioned: hr (good for regions with large stress gradients), hp or hpr
A nice mnemonic for these methods:
h-method = MesH
p-method = Polynomial order
r-method = Relocate node
- Check that the software you use is operating with the values you typed in!
- Look at the deformed shape in the post-processing part of your simulation and see if the result is roughly what you expected
Numerical Errors in FEA
- Integration error caused by Gauss integration leads to numerical instabilities. BUT a large number of Gauss points is way too expensive. For the basic concept of Gauss Quadrature see: Excursus – FEM – Gauss Quadrature
Useful information: Exact integrations provide a structure that is too stiff!2
- Rounding error caused by adding or subtracting very large and very small numbers or dividing by small numbers
- Matrix conditioning errors3
Practice makes perfect!
“Science walks forward on two feet, namely theory and experiment… but continuous progress is only made by the use of both.” – Robert A. Millikan
Want to read about practical applications of FEA? This case study shows a stress analysis of a wheel loader arm performed with the SimScale simulation platform.
 – Wikipedia – Boundary value problem (https://en.wikipedia.org/wiki/Boundary_value_problem)
 – Introduction to the Finite Element Method – Niels Ottosen & Hans Petersson- Chapter 20.3